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Mathematical Model of the Acute Inflammatory Response to Escherichia coli in Intramammary Challenge

J. Detilleux^*,1, F. Vangroenweghe and C. Burvenich

^* Faculty of Veterinary Medicine, University of Liège, Department of Quantitative Genetics, Liege 4000, Belgium
Milk Secretion and Mastitis Research Center, Department of Physiology and Biometrics, Faculty of Veterinary Medicine, Ghent University, Salisburylaan 133-9820 Merelbeke, Belgium

¹ Corresponding author: jdetilleux{at}ulg.ac.be

We constructed a mathematical model of the early response to Escherichia coli infection of the mammary gland and explored the roles and interactions between inflammatory cells and bacteria. The model incorporates 3 equations that describe the interactions among bacteria, milk somatic cells, and blood leukocyte densities. These 3 equations were fitted to cell densities observed during acute inflammatory responses in unvaccinated and vaccinated heifers inoculated with 10⁴ or 10⁶ cfu of E. coli. The rates computed for the cellular transit from the storage sites to the blood and from the blood to the milk were lower in cows receiving 10⁴ cfu but increased at approximately 6 x 10^–6 and 30 x 10^–6 µL/cfu per h in nonvaccinated or vaccinated cows inoculated with 10⁶ cfu, respectively. The cellular rates of bacterial killing were highest in unvaccinated cows (400 x 10^–6 µL/cell per h) when compared with vaccinated cows (200 to 300 x 10^–6 µL/cell per h). A critical density of milk somatic cells at which bacteria density is constant was computed from the model at 2 x 10⁶ cells/mL, and a one-way sensitivity analysis revealed that the changes in milk cellular densities were mostly sensitive to variations in the rate of bacterial killing and in the rate of production of inflammatory cells.

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